Electric current is the rate of flow of electric charges across any cross-sectional area of a conductor. The direction of electric current is taken as the direction of flow of positive ions or opposite to the direction of flow of free electrons. Your assumption is not necessary here... Electrons always flow from negative terminal to positive terminal.
When current flows through an electrolytic solution or during the process of electrolysis, The plate towards which positive ions (cations) flow is called the cathode and the plate towards which negative ions (anions) flow is called the anode.
Wikipedia says clearly,
In an electrochemical cell, The electrode at which electrons leave the cell and oxidation occurs is called anode and the electrode at which electrons enter the cell and reduction occurs is called cathode. Each electrode may become either the anode or the cathode depending on the direction of current through the cell. A bipolar electrode is an electrode that functions as the anode of one cell and the cathode of another cell.
So, the convention is totally based on our definition of the direction of current flow that it always flows opposite to the direction of electrons (i.e) electrons can be called as cations or anions depending on the usage. And based on this, we dump our thought that cathode should always be negative, etc...